All Pairs Shortest Distances for Graphs with Small Integer Length Edges

Galil, Zvi; Margalit, Oded

doi:10.1006/inco.1997.2620

Cited by 53 publications

(38 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The results of Seidel [Sei95] and Galil and Margalit [GM97a], [GM97b] show that the complexity of the APSP problem for unweighted undirected graphs isÕ(n ω ). The exact complexity of the directed version of the problem is not known yet.…”

Section: Discussionmentioning

confidence: 99%

“…It is fairly easy to see that paths of stretch 1 + ǫ between all pairs of vertices of an unweighted directed graph can be computed inÕ(n ω /ǫ) time. (This fact is mentioned in [GM97a]). Stretch 2 paths, or at least stretch 2 distances, for example, may be obtained by computing the matrices A 2 r , for 1 ≤ r ≤ ⌈log 2 n⌉, where A is the adjacency matrix of the graph, and Boolean products are used this time.…”

Section: Introductionmentioning

confidence: 93%

“…They obtained an algorithm whose running time isÕ(n (3+ω)/2 ) 1 for solving the APSP problem for directed graphs with edge weights taken from the set {−1, 0, 1}. Galil and Margalit [GM97a], [GM97b] and Seidel [Sei95], obtainedÕ(n ω ) time algorithms for solving the APSP problem for unweighted undirected graphs. Seidel's algorithm is much simpler.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

All pairs shortest paths using bridging sets and rectangular matrix multiplication

Zwick

2002

J. ACM

266

364

View full text Add to dashboard Cite

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms.The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value inÕ(n 2+µ ) time, where µ satisfies the equation ω(1, µ, 1) = 1 + 2µ and ω(1, µ, 1) is the exponent of the multiplication of an n × n µ matrix by an n µ × n matrix. Currently, the best available bounds on ω(1, µ, 1), obtained by Coppersmith, imply that µ < 0.575. The running time of our algorithm is therefore O(n 2.575 ). Our algorithm improves on theÕ(n (3+ω)/2 ) time algorithm, where ω = ω(1, 1, 1) < 2.376 is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be O(n 2.688 ).The second algorithm solves the APSP problem almost exactly for directed graphs with arbitrary nonnegative real weights. The algorithm runs inÕ((n ω /ǫ) log(W/ǫ)) time, where ǫ > 0 is an error parameter and W is the largest edge weight in the graph, after the edge weights are scaled so that the smallest non-zero edge weight in the graph is 1. It returns estimates of all the distances in the graph with a stretch of at most 1 + ǫ. Corresponding paths can also be found efficiently.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

All pairs shortest paths using bridging sets and rectangular matrix multiplication

Zwick

2002

J. ACM

266

364

View full text Add to dashboard Cite

show abstract

“…In fact, if ω = 2 then both the algorithms of [3] and [24] run in O(n 2.5 ), so if ω = 2 then the last progress on the APSP problem in directed graph with small weights (or even unweighted) was [3] from 1991. We finally note that the only (general) case where the APSP problem can be solved in time O(n ω ) is in the case of undirected graphs with small weights, see [15,16,21,20].…”

Section: O(n ω+3mentioning

confidence: 99%

All-pairs shortest paths with a sublinear additive error

Roditty

Shapira

2011

ACM Trans. Algorithms

View full text Add to dashboard Cite

We show that for every 0 ≤ p ≤ 1 there is an O(n 2.575−p/(7.4−2.3p) ) time algorithm that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u, v in the graph to within an additive error δ p (u, v), where δ(u, v) is the exact length of the shortest path between u and v. This algorithm runs faster than the fastest algorithm for computing exact shortest paths for any 0 < p ≤ 1.Previously the only way to "bit" the running time of the exact shortest path algorithms was by applying an algorithm of Zwick [FOCS '98] that approximates the shortest path distances within a multiplicative error of (1 + ). Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error. In fact, the main ingredient we need in order to obtain the above result, which is also interesting in its own right, is an algorithm for computing (1 + ) multiplicative approximations for the shortest paths, whose running time is faster than the running time of Zwick's approximation algorithm when 1 and the graph has small integer weights.

show abstract

“…Galil et al [9] have improved the dependence on W and have also given an O(W (ω+1)/2 n ω log n) algorithm for undirected graphs. Ravi et al [18] and Mirchandani [14] have given sequential algorithms to solve APSP problem on an interval graph in O(n 2 ) time.…”

Section: Introductionmentioning

confidence: 99%

An optimal parallel algorithm for solving all-pairs shortest paths problem on circular-arc graphs

Saha

Pal

2005

JAMC

View full text Add to dashboard Cite

Abstract. The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents an O(n 2 ) time sequential algorithm and an O(n 2 /p + log n) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, where p and n represent respectively the number of processors and the number of vertices of the circular-arc graph.

show abstract

All Pairs Shortest Distances for Graphs with Small Integer Length Edges

Cited by 53 publications

References 1 publication

All pairs shortest paths using bridging sets and rectangular matrix multiplication

All pairs shortest paths using bridging sets and rectangular matrix multiplication

All-pairs shortest paths with a sublinear additive error

An optimal parallel algorithm for solving all-pairs shortest paths problem on circular-arc graphs

Contact Info

Product

Resources

About